Dispersive Estimates for Schrödinger Operators in Dimensions One and Three

We consider L¹L estimates for the time evolution of Hamiltonians H=–+V in dimensions d=1 and d=3 with bound We require decay of the potentials but no regularity. In d=1 the decay assumption is (1+|x|)|V(x)|dx<, whereas in d=3 it is |V(x)|C(1+|x|)^–3–.Supported by the NSF grant DMS-0070538 and a Sloan fellowship.     

Dispersive Estimates for Schrödinger Operators in Dimension Two

We prove L¹(²)L(²) for the two-dimensional Schrödinger operator –+V with the decay rate t^–1. We assume that zero energy is neither an eigenvalue nor a resonance. This condition is formulated as in the recent paper by Jensen and Nenciu on threshold expansions for the two-dimensional resolvent.The author was partially supported by the NSF grant DMS-0300081 and a Sloan Fellowship     

A Weighted Dispersive Estimate for Schrödinger Operators in Dimension Two

Let H = ?Δ + V, where V is a real valued potential on ${\mathbb {R}^2}$ satisfying ${\|V(x)|\lesssim \langle x \rangle^{-3-}}$ . We prove that if zero is a regular point of the spectrum of H = ?Δ + V, then $${\| w^{-1} e^{itH}P_{ac}f\|_{L^\infty(\mathbb{R}^2)} \lesssim \frac{1}{|t|\log^2(|t|)} \| w f\|_{L^1(\mathbb{R}^2)},\,\,\,\,\,\,\,\, |t| \geq 2}$$ , with w(x) = (log(2 + |x|))². This decay rate was obtained by Murata in the setting of weighted L ² spaces with polynomially growing weights.     

Tunneling Estimates for Magnetic Schrödinger Operators

We study the behavior of eigenfunctions in the semiclassical limit for Schr?dinger operators with a simple well potential and a (non-zero) constant magnetic field. We prove an exponential decay estimate on the low-lying eigenfunctions, where the exponent depends explicitly on the magnetic field strength. Received: 30 March 1998 / Accepted: 1 May 1998     

Eigenvalue Estimates for Schrödinger Operators with Complex Potentials

We discuss properties of eigenvalues of non-self-adjoint Schrödinger operators with complex-valued potential V. Among our results are estimates of the sum of powers of imaginary parts of eigenvalues by the L ^p -norm of \({{\Im{V}}}\).     

A Liouville-type Theorem for Schrödinger Operators

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P ₁, such that a nonzero subsolution of a symmetric nonnegative operator P ₀ is a ground state. Particularly, if P _j : = ?Δ + V _j , for j = 0,1, are two nonnegative Schrödinger operators defined on \(\Omega\subseteq \mathbb{R}^d\) such that P ₁ is critical in Ω with a ground state φ, the function \(\psi\nleq 0\) is a subsolution of the equation P ₀ u = 0 in Ω and satisfies \(\psi_+\leq C\varphi\) in Ω, then P ₀ is critical in Ω and \(\psi\) is its ground state. In particular, \(\psi\) is (up to a multiplicative constant) the unique positive supersolution of the equation P ₀ u = 0 in Ω. Similar results hold for general symmetric operators, and also on Riemannian manifolds.     

Estimates for Eigenvalues of Schrödinger Operators with Complex-Valued Potentials

New estimates for eigenvalues of non-self-adjoint multi-dimensional Schrödinger operators are obtained in terms of L_p-norms of the potentials. The results cover and improve those known previously, in particular, due to Frank (Bull Lond Math Soc 43(4):745–750, 2011), Safronov (Proc Am Math Soc 138(6):2107–2112, 2010), Laptev and Safronov (Commun Math Phys 292(1):29–54, 2009). We mention the estimations of the eigenvalues situated in the strip around the real axis (in particular, the essential spectrum). The method applied for this case involves the unitary group generated by the Laplacian. The results are extended to the more general case of polyharmonic operators. Schrödinger operators with slowly decaying potentials and belonging to weak Lebesgue’s classes are also considered.     

Unique Continuation Principle for Spectral Projections of Schrödinger Operators and Optimal Wegner Estimates for Non-ergodic Random Schrödinger Operators

We prove a unique continuation principle for spectral projections of Schrödinger operators. We consider a Schrödinger operator H = ?Δ + V on ${{\rm L}^2(\mathbb{R}^d)}$ L 2 ( R d ) , and let H _Λ denote its restriction to a finite box Λ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the type χ _I (H _Λ ) W χ _I (H _Λ ) ≥ κ χ _I (H _Λ ) with κ > 0 for appropriate potentials W ≥ 0 and intervals I. As an application, we obtain optimal Wegner estimates at all energies for a class of non-ergodic random Schrödinger operators with alloy-type random potentials (‘crooked’ Anderson Hamiltonians). We also prove optimal Wegner estimates at the bottom of the spectrum with the expected dependence on the disorder (the Wegner estimate improves as the disorder increases), a new result even for the usual (ergodic) Anderson Hamiltonian. These estimates are applied to prove localization at high disorder for Anderson Hamiltonians in a fixed interval at the bottom of the spectrum.     

Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue

Let H=−Δ+V(x) be a three dimensional Schrödinger operator. We study the time decay in L^p spaces of scattering solutions e^−itHP_cu, where P_c is the orthogonal projection onto the continuous spectral subspace of L²(R³) for H. Under suitable decay assumptions on V(x) it is shown that they satisfy the so-called L^p-L^q estimates ||e^−itHP_cu||_p≤(4π|t|)^{−3(1/2−1/p)}||u||_q for all 1≤q≤2≤p≤∞ with 1/p+1/q=1 if H has no threshold resonance and eigenvalue; and for all 3/2<q≤2≤p<3 if otherwise.     

Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign

Let H be a one-dimensional discrete Schrödinger operator. We prove that if _ess(H)[–2,2], then H–H₀ is compact and _ess(H)=[–2,2]. We also prove that if has at least one bound state, then the same is true for H₀+V. Further, if has infinitely many bound states, then so does H₀+V. Consequences include the fact that for decaying potential V with , H₀+V has infinitely many bound states; the signs of V are irrelevant. Higher-dimensional analogues are also discussed. Supported in part by NSF grant DMS-0227289On leave from Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801-2975, USASupported in part by NSF grant DMS-0140592     

Inverse Spectral Problems for Schrödinger Operators

In this article we find some explicit formulas for the semi-classical wave invariants at the bottom of the well of a Schrödinger operator. As an application of these new formulas for the wave invariants, we improve the inverse spectral results proved by Guillemin and Uribe in [GU]. They proved that under some symmetry assumptions on the potential V(x), the Taylor expansion of V(x) near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schrödinger operator in \({\mathbb R^n}\) . We prove similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of V(x).     

Schr?dinger Dispersive Estimates for a Scaling-Critical Class of Potentials

We prove a dispersive estimate for the evolution of Schr?dinger operators H = ??? + V(x) in ${{\mathbb R}^3}$ . The potential should belong to the closure of ${C^c_b(\mathbb{R}^3)}$ with respect to the global Kato norm. Some additional spectral conditions are imposed, namely that no resonances or eigenfunctions of H exist anywhere within the interval [0, ??). The proof is an application of a new version of Wiener??s L ¹-inversion theorem.     

Generic Continuous Spectrum for Ergodic Schrödinger Operators

We consider families of discrete Schrödinger operators on the line with potentials generated by a homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon’s Lemma that for a generic continuous sampling function, the set of elements in the associated family of Schrödinger operators that have no eigenvalues is large in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.     

A New Estimate for the Groundstate Energy of Schrödinger Operators

A new estimate for the groundstate energy of Schrödinger operators on L²(ⁿ) (n 1) is presented. As a corollary, it is shown that the groundstate energy of the Schrödinger operator with a scalar potential V is more than the classical lower bound ess.inf_x__V(x) if V is essentially bounded from below in a certain manner (enhancement of the groundstate energy due to quantization). As an application, it is proven that the groundstate energy of the Hamiltonian of the hydrogen-like atom is enhanced under a class of perturbations given by scalar potentials (vanishing at infinity).     

Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains

On a smooth bounded domain \(\Omega \subset {\bf {\rm R}}^N\) we consider the Schrödinger operators ? Δ ? V, with V being either the critical borderline potential V(x) =  (N ? 2)²/4 |x|^?2 or V(x) =  (1/4) dist(x, ?Ω)^?2, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Schrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a series of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincaré. As a byproduct of our technique we are able to answer positively to a conjecture of E. B. Davies.     

On the Inverse Resonance Problem for Schrödinger Operators

We consider Schrödinger operators on [0, ∞) with compactly supported, possibly complex-valued potentials in L ¹([0, ∞)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data. In this paper we prove the validity of this statement, i.e., we show conditional stability for finite data. As a by-product we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials.     

Perturbations of Magnetic Schrödinger Operators

We study Schrödinger operators H(a, V): = (P – a)² + V acting in L ²(³). We assume that the magnetic field B = rot a may be decomposed as B = B ⁰ + B, where B ⁰ is a very general field having constant direction. The perturbations B and V will be small in a certain sense in the direction of B ⁰, but in the orthogonal plane they may even grow for certain fields B ⁰. Commutator methods are used to derive spectral properties of H(a, V).